Crunching the numbers for nuclear fusion

Ah, fusion. Long promised, both on Do the Math and in real life, fusion is regarded as the ultimate power source — the holy grail — the “arrival” of the human species. Talk of fusion conjures visions of green fields and rainbows and bunny rabbits . . . and a unicorn too, I hear. But I strike too harsh a tone in my jest.

Fusion is indeed a stunningly potent source of energy that falls firmly on the reality side of the science fiction divide — unlike unicorns. Indeed, fusion has been achieved (sub break-even) in the lab, and in the deadliest of bombs. On the flip side, fusion has been actively pursued as the heir-apparent of nuclear fission for over 60 years. We are still decades away from realizing the dream, causing many to wonder exactly what kind of “dream” this is.

Our so-far dashed expectations seem incompatible with our sense of progress. Someone born in 1890 would have seen horses give way to cars, airplanes take to the skies, the invention of radio, television, and computers, development of nuclear fission, and even humans walking on the Moon by the age of 79. Anyone can extrapolate a trajectory, and this trajectory intoned that fusion would arrive any day — along with colonies on Mars. Yet we can no longer buy a ticket to cross the Atlantic at supersonic speeds, and the U.S. does not have a human space launch capability any more. Even so, fusion remains “just around the corner” in many minds.

I am sympathetic to delayed predictions, and the fact that fusion has failed to deliver on the promise that it’s “just around the corner” for decades does not mean that it will never arrive. I can compare this to Malthus’ insight that exponential population growth was on a collision course with finite agricultural capability, or to various warnings about collapse along the way. Just because the predictions have not yet been satisfied does not mean that they will not be someday. In fact, the two divergent predictions become related. If we can manage to hold it together this century and maintain a high-tech civilization during our forced transition off of fossil fuels, it becomes far more likely that we will get to the point of employing fusion. If, on the other hand, we overshoot and collapse, we may descend too far to viably pursue fusion this century.

Fusion by the Numbers

What’s fusion all about, anyhow? Let’s come at it with numbers. We saw in the post on nuclear fission that allowing a heavy nucleus like uranium to split into two comparable pieces resulted in the sum of the resultant masses being less than the initial mass. The missing mass emerges as (kinetic) energy according to E = Δmc², where Δm is the change in mass, and c ≈ 3×108 m/s is the speed of light. In essence, some of the nuclear binding energy invested the heavy nucleus — which actually reduces the net mass of the nucleus — has been liberated.

To understand this better, consider the fact that a single neutron has a mass of 1.08665 atomic mass units (amu: 1.66×10−27 kg), and a neutral hydrogen atom (one proton plus one electron, minus a trivial amount of electromagnetic binding energy: just 14 parts per billion) has a mass of 1.007825 amu. To make 235U, we take 92 hydrogen atoms, add 143 neutrons, and stir. Without considering nuclear binding energy, the sum would be 236.96 amu. Yet the neutral 235U atom has a mass of 235.044 amu. The “missing” 1.92 amu is the nuclear energy that would be released by building (fusing) this ensemble.

Think of it this way: when a nucleus grabs hold of a passing neutron, the deathly-strong nuclear grip slams the neutron into the nucleus, momentarily giving it kinetic energy. Initially, the nucleus jiggles like jello in an excited state, before releasing this energy (via gamma ray, or fast electron in beta decay, etc.) back to the world. In releasing this energy, its mass must decrement in deference to Einstein’s most famous relation. In this way, every nucleon added (proton or neutron) contributes its direct mass to the nucleus, but then subtracts about 0.008 amu of binding energy, on average — in effect weighing in at only 0.992 amu-a-pop.

Of fundamental importance in appreciating the energy gains inherent in fusion and fission processes is the chart of binding energy per nucleon. The graph below plots the binding energy per nucleon in units of MeV, where 1 MeV = 1.6×10−13 J and is equivalent to 0.00107 amu via E = mc². Or, roughly speaking, 1 MeV is one-thousandth the mass of a single nucleon. The horizontal axis of the plot is the total number of nucleons—protons plus neutrons—in the nucleus.

Higher binding energy translates to smaller net mass, compared to the dumb sum of constituent masses. So the higher on the curve, the more energy can be given up in building that nucleus. Iron sits at the top (with plenty of company in neighbors like nickel). On the left side, adding pieces together constitutes a net energy gain (fusion), while on the right, one must tear nuclei apart (fission) to climb up the hill. Thus it is said that fusion yields net energy for atoms smaller than iron, and that fission yields energy for atoms heavier than iron.

But let’s refine that point. If I tried to split 86Kr, for instance, at 8.71 MeV/nuc into two 43Ca atoms at 8.60 MeV/nuc, I have not climbed up the binding energy hill. In practice, one must have mass number above about 100 before fission into two equal pieces will release net energy. But the point is almost meaningless, given that the only three nuclei susceptible to slow-neutron fission have 233, 235, and 239 nuclei — well above the threshold for energy gain.

You may have noticed by now that if climbing the hill is the goal for energy gain, we have a lot more climb available on the left (fusion) side than on the right (fission) side. In particular, notice 4He sitting pretty atop a local spike. 4He is such a tightly-bound nucleus that heavy nuclei undergoing radioactive decay often eject one of these hard nuggets like a boxer spitting out a tooth, called alpha decay. 238U, for instance, will typically spit out 8 “teeth” and 6 electrons (beta) in its journey to become 206Pb. In any case, 4He is unique among nuclei, and bears the special name of alpha particle.

For example, building a 4He nucleus out of four protons—as our Sun is so talented at doing—we gain 28.3 MeV (7.07 MeV/nuc times four nucleons). Second-best would be starting with two deuterium (2H, or D) nuclei to build 4He. In this case, we go from two nuclei bound at 1.112 MeV/nuc (times two nucleons each; then times two deuterons for 4.45 MeV total) to 28.3 MeV for a total climb of 23.85 MeV. Still pretty darned good: not much penalty starting with D. Another relevant starting point is combining D with tritium (3H, or T), popping out the unwanted neutron. In this case, we start at 7.88 MeV total, for a net climb of 20.4 MeV.

Compared to fission, where each split releases about 200 MeV of energy, it might appear that this fusion stuff is comparatively wimpy — seeming out of kilter when we look at the steeper slope for fusion on the binding energy plot. The discrepancy is the number of nucleons involved. Mirroring the example in the nuclear fission post, 235U, at 7.6 MeV/nuc splits into 97Rb and 137Cs at about 8.4 MeV/nuc each. Although the slope is meager (a mere 0.8 MeV/nuc step), multiplying by the nucleon number yields a binding energy gain of 97×8.4 + 137×8.4 − 235×7.6 = 180 MeV.

On a per mass, or per nucleon basis, fusion wins hands-down: one gram of deuterium results in 1012 J of energy, or 275 million kcal. Fission gives a comparatively small 20 million kcal per gram of 235U. So fusion is over ten times as potent. Keep in mind that chemical energy like that in fossil fuels is capped around 10 kcal/g. Note the conspicuous absence of the word million. On the energy scale, then, nuclear in either form is outrageously more potent than chemical energy.

Fusion Fuel Options

The two fusion schemes for which we can produce the requisite fuel are D-D and D-T, involving deuterium and/or tritium. Deuterium comprises 0.0115 percent of natural hydrogen, and is thus abundant in anything containing hydrogen — e.g., water. Tritium, on the other hand, is virtually non-existent in the natural world because it is unstable and decays with a half-life of 12.3 years. But as it happens, the requirements on D-T fusion are less impossible than for D-D, so all current efforts are focused on a technique for which there is no natural resource available.

Okay, so the pointy-heads aren’t that stupid. There is a way to create 3H by smacking lithium (either 6Li or 7Li) with a neutron and knocking out a tooth—er, 4He—leaving either 3H or 4H (in the latter case promptly dripping a neutron to become tritium).

I find it helpful to consult a chart of the nuclides when considering such shenanigans. Here is the bottom-end of the chart, which is basically the physicist’s version of a periodic table.

The number of neutrons increases from left to right, and the number of protons increases vertically. Thus all helium nuclei will be on the same row, for instance. Gray shading indicates a stable nucleus (stable well beyond the age of the Universe), light blue is semi-stable, and yellow less so. Each block contains the name of the nucleus/isotope, the fractional abundance (if stable), the half life (if unstable), the mass of the neutral atom in atomic mass units, and the decay path (arrows). Decays can be beta-minus (blue, transition to upper left), beta-plus (magenta to lower right), alpha (long yellow arrow to lower left), neutron drip (green arrow to left), or proton drip (red arrow down) These are the chess-board rules. Incidentally, it is possible to reconstruct binding energies from the mass numbers in each block.

We can use the chart to follow the two reaction types:

D + D → 4He

The D-D reaction is pretty straightforward. Marrying two nuclei together, each with one proton and one neutron, the result has two protons and two neutrons. No extra neutrons are generated in the bargain.

For D-T, we must first create the tritium from either flavor of lithium:

6Li + n → 4He + T, or

7Li + n → 4He + 4H → 4He + T + n

In either case, the “decay” chain is not the natural one, but is jarred out of the nucleus in the impact. Nominally, adding a neutron to 6Li just yields the stable 7Li, and adding a neutron to 7Li makes 8Li, which beta-decays in about a second to 8Be and then instantly splits into two alpha particles (4He). But in smackdown mode, one can conjure tritium, possibly yielding an extra neutron, depending on the isotope of lithium used. Then we have:

D + T → 5He → 4He + n

Note the extra neutron. This is handy, since we need neutrons to convert lithium to tritium. But note also that using 7Li generates two neutrons per D-T reaction, while 6Li only generates the one. Neutrons will be lost to other parasitic causes, so it’s handy to have extras around. On the other hand, neutron capture by the containment vessel makes it radioactive and will also damage its structural integrity, so we want to be careful about how many extra neutrons there are. Unfortunately, natural lithium is 92.4% 7Li, so tuning the 6Li/7Li mix to give the critical number of neutrons implies some sort of lithium enrichment on the front-end.

We aren’t exactly swimming in lithium, so did we make a bad trade in picking this horse? Each lithium atom converted to tritium will end up yielding about 20 MeV of thermal energy, so that we need 1.3×1032 Li atoms annually to produce our world consumption of 4×1020 J. That’s about 1500 metric tons of lithium annually, or about 5% of current lithium production. Proven world reserves give us 9000 years, and estimated resources give us 22,000, according to the U.S.G.S. Mineral Commodities Summaries.

For fun, let’s look at how much water each person needs to supply each year to provide enough deuterium. The average American demands 10,000 W of continuous power, or 3×1011 J of energy per year. At 20 MeV per whack, each person needs 1023 reactions per year. In the D-D case (requiring twice the deuterium as D-T), this means we need 2×1023 deuterium atoms—coming from 2×1027 hydrogen atoms at a fractional abundance of 0.01 percent. Sounds like a lot, but it’s 3,300 moles — amounting to 60 kg of ordinary water. 60 liters is similar to the amount of water used in a typical American shower. It’s hard to emphasize enough the extent to which deuterium availability poses no problem: there is enough deuterium in the ocean to provide our current energy demand for billions of years.

I think now you’re seeing a big part of the reason why fusion makes our eyes sparkle. Even given lithium limitations, I place D-D and D-T fusion in the “abundant” box.

What Makes Fusion Hard

A simple obstacle stands between us and fusion. It’s called the Coulomb barrier. Protons hate to get near each other, on account of their mutual positive charge and concomitant electrostatic repulsion. And they must get very close—about 10−15 m—before the strong nuclear force overpowers Coulomb’s vote. Even on a perfect collision course, two protons would have to have a closing velocity of 20 million meters per second (7 percent the speed of light) to get within 10−15 m of each other, corresponding to a temperature around 5 billion degrees! Even if the velocity is sufficient, the slightest misalignment will cause the repulsive duo to veer off course, not even flirting with contact. Quantum tunneling can take a bit of the edge off, requiring maybe a factor of two less energy/closeness, but all the same, it’s frickin’ hard to get protons together.

Yet our Sun manages to do it, at a mere 16 million degrees in its core. How does it manage to make a profit? Volume. The protons in the Sun are racing around at a variety of velocities according to the temperature. While the typical velocity is far too small to defeat the Coulomb barrier, some speed demons on the tail of the velocity distribution curve do have the requisite energy. And there are enough of them in the vast volume of the Sun’s core to occasionally hit head on and latch together. One of the protons must promptly beta-plus decay into a neutron and presto-mundo, we have a deuteron!

Deuterons can then collide to make helium (other paths to helium are also followed). A quick and crude calculation suggests that we need about 1038 “sticky” collisions per second to keep the Sun going, while within the core we get about 1064 bumps/interactions per second, implying only one in 1026 collisions needs to be a successful fusion event.

Deuterons have an easier time bumping into each other than do lone protons, mainly because their physical size is larger. In fact, a deuteron’s relatively weak binding makes them even puffier than the more tightly bound tritium nucleus (go tritons!). At a given temperature, deterons will move more slowly than protons, and tritons more slowly than deuterons. All flavors contain a single proton—and so exert the same repulsive force on each other—but the increased inertia from extra neutrons exactly counters the slower speed, so that each has the same likelihood of trucking through the Coulomb barrier. Then we’re left with size. Deuterons are bigger than tritons, so D-D bumps will be more common than D-T bumps.

But there’s a catch. As soon as D and T touch, they stick together. Conversely, when D touches D, a photon (light) must be emitted in order for them to stick, which doesn’t usually happen. It is therefore said that D-T has a greater cross section for fusion than D-D. Estimates for the critical temperature required to achieve fusion come in at 400 million Kelvin for D-D fusion, and 45 million K for the D-T variety. But these temperature thresholds depend on the density of the plasma involved, so should not be taken as hard-and-fast. Still, we need our fusion reactors to be hotter than the center of the Sun because we do not have the luxury of volume and density that the solar core enjoys. Does this fact give you pause?

Confinement

Overcoming the Coulomb barrier requires enormous kinetic energies of the particles, translating into enormous temperatures — well beyond any container’s ability to hold. No material resists melting above a mere 5000 K. 50 million degrees is not even funny.

At these temperatures/energies, electrons are not able to hold onto their rides, so we get a completely ionized plasma zipping this way and that. At 100 million degrees, for instance, deuterium nuclei have an average velocity of about one million meters per second. Left alone, the plasma would explode to the size of a football field in 0.1 milliseconds. Recall that we can’t get fusion to happen without these ridiculous velocities, so we’re stuck having to herd these hyper-fast particles without the help of Ritalin. It has been found that plasmas at the requisite temperature suffer instabilities from turbulence that we have been unable to tame. It becomes like a game of whack-a-mole, according to my colleague George Fuller: clamp down on one pesky behavior, and another one pops up.

The main scheme being pursued in the world today is magnetic confinement in a plasma containment vessel called a tokamak. Charged particles follow curved arcs in a magnetic field, so that strong fields confine the particle paths to tight curls. The radius of the path is proportional to the particle velocity, which spans a large range of values in a thermal plasma. One must produce a magnetic field strong enough to contain the fast tail of the velocity distribution, else the plasma has a leak at the high-velocity end and depletes itself rather quickly. Every particle collision resets velocities, so a leaking fast tail is constantly re-populated. At a field strength of 10 Tesla (near the upper end achievable), the mean-velocity deuteron at 50 million K has a 2 mm path radius. ITER, the International Thermonuclear Experimental Reactor, is a tokamak design being built in France under international support. The current timeline calls for achievement of a 480 second burst of 500 MW power in the year 2026, although there is no plan to capture the generated heat for the production of electricity (note the “Experimental” in the project name).

The other primary scheme gives up on trying to confine the plasma in some steady state, instead following a path similar to the philosophy behind fusion bombs: force an implosion of the fuel to extraordinarily high densities and temperatures, and let the cursed thing explode. This scheme goes under the name inertial confinement, since one relies on the inertia of the implosion to bring nuclei close together. In the U.S., the National Ignition Facility (NIF) focuses 192 high-power laser beams onto a small pellet to initiate a symmetric crunch. The idea for a power plant would be that pellets are loaded one after the other, detonated, and the effluent heat collected to make steam. As far as I know, there is no current plan to harness any heat generated at the NIF—being experimental, like ITER.

Flies in the Ointment

The ITER experiment, if it adheres to its schedule and projected budget, will cost something like $20 billion to build and produce pops of unharnessed thermal power by 2026. I should note that most large experimental projects have slipping schedules, and it would be a fantastic irony if a fusion experiment violated this trend! In any case, we could imagine another several decades before commercial fusion tentatively steps onto the scene, putting us at mid-century. The projects will undoubtedly be very expensive, require intimate involvement of the highest level of expertise, and will likely not catch on in a big way until investors see a track record of profitability—if that ever comes to pass. So that’s fly number one: we’re looking at very long term.

Fly number two is that D-T fusion necessarily involves neutrons, which do not respond to magnetic or electrostatic confinement and therefore hurtle off to the walls of the containment vessel. In doing so, they knock into the atoms comprising the vessel, dislocating them within the lattice and causing structural damage. The integrity of the containment vessel will degrade like plastic in sunlight. The neutron flux from a D-T reactor is substantially higher than for a conventional fission reactor.

Fly number three is also related to neutrons: after doing their damage in the containment walls, the neutrons will marry a nice, plump nucleus and settle down. But the marriage is often radioactive, so that the container becomes radioactively “hot.” In fission, we get two radioactive daughters for each 200 MeV produced. For D-T fusion, if we are able to utilize most of the neutrons for conversion of lithium into tritium (and use enriched 6Li), we might be able to lose less than 0.2 neutrons per 20 MeV reaction (pure, uninformed guess on my part), which comes out to the same number of radioactive products per unit of energy. But at least materials choices for the container walls offers some control over the menagerie of radioactive products—unlike the randomness of fission. All told, the radioactive toll from a D-T fusion reactor may be comparable to that of a fission reactor, though with shorter half-life.

Then there is the extremely finicky nature of achieving fusion. Getting something to work in the lab is much different from having it operate reliably for years on end. Any significant departure from optimal conditions will see the fusion yield diminish. ITER aims for a thermal output ten times that of the input energy. In an eventual self-running mode, siphoning 10% of the output power in electrical form requires pulling out about 30% of the thermal power to run the heat-engine generator. This makes for a 3:1 net energy gain, which could quickly transition to a net energy drain if things are not maintained in tip-top condition through the years.

Another possible fly is that the superconducting magnets used to generate the extreme magnetic fields for confinement could lose cryogenic cooling, “go normal,” and explode. An explosion that damaged the tokamak could result in a radioactive release to the environment. Even though the probability is small, we routinely go to great expense to mitigate low-probability catastrophic events, and so a massive, expensive containment building would likely be required.

Each fly translates into cost. In the end, it is unclear whether a fusion plant—even after the physics is tamed—would be economically viable, and attractive enough for investors to take on endeavors of this scale, complexity, and risk.

A Solar Perspective

A few days after watching a television show on fusion, I had an epiphany while walking to the bus. Why are we enamored with fusion? Because the fuel supply is virtually unlimited; the energetics represent the epitome of what physics has to offer; the primary emission is useful helium; the radioactive waste is shorter-lived than for fission (damning with faint praise?); fusion plants could presumably be sited anywhere; surely it’s one step closer to warp drive. But then I realized that the Sun (being its own fusion reactor) also provides billions of years of energy, well in excess of our current demand. And my refrigerator and other appliances already are run by this source in a modest PV/battery installation at my home. I personally can’t ignore the asymmetry between the promise of future technology and technology that sits on my roof! If we removed the storage barrier for solar, would fusion still be viewed as the holy grail?

This prompts two questions. First, what is the relative funding expenditure for fusion research and for battery/storage research? Second, what are the appeals offered by fusion that could leave solar in the shade?

A cursory investigation reveals that the U.S. spends approximately $450M per year on the NIF, and chips in about $32M per year to ITER (though expected to escalate to about $350M/year during the construction phase from 2014–2016). Meanwhile, the U.S. Department of Energy Hub for Batteries and Energy Storage plans to operate at $24M per year, with a similar expenditure in Fuels from Sunlight. It’s about as I thought.

I can only muse about the appeals of fusion over solar. I think area is one: fusion plants could be comparatively compact. I think location-dependence is another. Most people don’t realize that the worst site in the continental U.S. (Olympic peninsula) delivers fully half as much annual solar energy as the Mojave desert. Given a good storage solution, solar becomes useful almost anywhere. I think in part, we are driven by the sense of progress/conquest. Cracking the fusion problem matches our precious narrative. But I am left wondering if these reasons are compelling enough to keep us reaching for the gold that may continue to disappoint when we have other options whose viability may be closer at hand.

Naturally, it’s not an all-or-nothing proposition. I support research whatever the direction. But I want to make sure we aren’t falling victim to irrational hangups and expectations. We at least need to evaluate this notion: to know ourselves. One may object that I’ve simply replaced one holy grail (fusion) for another (storage). Which one is voted more likely to succeed?

Fusion Prospects

No one can truly say whether we will achieve fusion in a way that is commercially practical. If teams of PhDs have spent over 60 years wailing on the problem while spending tens of billions of dollars, I think it’s safe to use our fusion quest as the definition of hard. It’s a much larger challenge than sending men to the Moon. We have no historical precedent for an arduous technological problem on this scale that ultimately succeeded to become a ho-hum commercial reality. But for that matter, I don’t think we have any precedent for something on this scale that has failed. In short, we’re out of our depths and can’t be cocky about predictions in either direction.

I am hopeful that fusion can one day become a practical reality. I certainly understand it to be feasible in principle. My misgivings mainly lie in the extreme complexity of the challenge. It may take a year of intense study to become an expert on a coal-fired plant, to the point of being a go-to resource for troubleshooting and maintenance. A nuclear fission plant may take five years to master—it took about that long to get the first break-even performance after discovery of fission. But after a century of development (by the time any commercial fusion reactor sees the light of day), how long must one study plasma physics in order to have a firm handle on operation of a fusion plant? The NIF uses two lasers occupying a floorspace the size of a Wal-mart store (no exaggeration). How many PhDs will it take to keep a state-of-the-art laser of this magnitude operating? I know that the 2 W laser I use in my research causes this PhD enough trouble!

I became interested in energy because I sensed that we are approaching a phase change in society as the age of fossil fuels begins to ebb. So much of what we have become can be attributed to cheap and abundant surplus energy. Our energy future is highly uncertain. Commercial fusion may come along decades down the road—mid-century at the earliest—but even then it is yet another source of heat that we can use to make electricity. Another step (mobile storage) must accompany fusion development to replace petroleum functions, and even then at significant disadvantage in energy density using current technologies. So yeah—I hope it helps us out one day. But I’m not sure we can wait that long.